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Determining the genus
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
These are subrings, and S = T if and only if S = T. They are integrally closed. The holomorphy rings are characterized among the subrings by the property of being integrally closed (Corollary 3.2.8 of [198]). Another important property is that S is a principal ideal domain if S is a finite set of places.
Canonical Forms
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Results about each type of canonical form are presented in the section on that canonical form, which facilitates locating a result, but obscures the connections underlying the derivations of the results. The facts about all of the canonical forms discussed in this section can be derived from results about modules over a principal ideal domain; such a module-theoretic treatment is typically presented in abstract algebra texts, such as [DF04, Chap. 12].
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
A principal ideal ring (or more specifically, a principal ideal domain (P.I.D.)) is a commutative ring A without zero divisors and with unit element 1 (that is, an integral domain) and in which every ideal is principal.
General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings
Published in International Journal of Mathematical Education in Science and Technology, 2020
A. Cuida, F. Laudano, E. Martinez-Moro
In this final section we will point to some known results for evaluating and dividing on skew polynomial rings. This will show the potential of the topic all around the mathematical instruction, note that also this approach has been used actually in applied mathematics as for example in coding theory (see for example Martínez-Penas, 2018). We will rectrict ourselves to skew polynomial rings over fields with no derivation but the same results can be stated easily for general skew polynomial rings over division rings (Lam & Leroy, 1988). Let F be a field and σ an automorphism of F. The skew polynomial ring is the set of polynomials over the field F where the addition is defined as the usual one in the polynomial ring and the multiplication is defined as follows Indeed it a non-commutative ring unless σ is the identity. One can see that the Euclidean algorithm holds for right division which implies that R is a left principal ideal domain (Lam & Leroy, 1988). So following the previous sections we can evaluate a (left) polynomial on . Note that the obviuous ‘evaluation’ is wrong since it does not take into account the action of σ, but if we define whe have the following Factor Theorem